Sample Space and Events — Mathematics Knowledge System

Overview

In probability theory, the sample space of an experiment or random trial is the set of all possible outcomes or results of that experiment. A sample space is usually denoted using set notation, and the possible ordered outcomes, or sample points, are listed as elements in the set. It is common to refer to a sample space by the labels S, Ω, or U. The elements of a sample space may be numbers, words, letters, or symbols, and can be finite, countably infinite, or uncountably infinite. An event is any subset of the sample space — a collection of outcomes we're interested in assigning a probability to.

Intuition

Before you can talk about the probability of anything, you first have to nail down exactly what outcomes are possible. Rolling a six-sided die, the sample space is {1,2,3,4,5,6} — every possible thing that could happen. 'Rolling an even number' isn't itself an outcome; it's a group of outcomes, {2,4,6}, carved out of the sample space. That grouping is what mathematicians call an event. Sample spaces can be small and countable (a coin flip: {H,T}), infinite but countable (the number of trials until a coin first lands heads: {1,2,3,...}), or uncountable (the exact arrival time of a bus, a continuum of real numbers).

Formal Definition

Definition

The sample space Ω of a random experiment is the set of all possible outcomes. An event A is any subset of Ω, i.e. A ⊆ Ω. The set of all events forms an event space (typically a σ-algebra) on which a probability measure P is defined, satisfying P(Ω) = 1 and P(∅) = 0.

\Omega = \{\omega_1, \omega_2, \ldots\}

The set of all possible outcomes of an experiment

Sample space

A \subseteq \Omega

An event is any subset of the sample space

Event

P(\Omega) = 1, \quad P(\varnothing) = 0

The whole sample space has probability 1; the impossible event has probability 0

Notation

Notation	Meaning
$\Omega$	The sample space (also written S or U)
$\omega$	A single outcome (sample point) in the sample space
$A \subseteq \Omega$	An event — a subset of the sample space
$A^c$	The complement of event A — all outcomes not in A
$A \cup B$	Event 'A or B occurs'
$A \cap B$	Event 'A and B both occur'

Properties

Complement rule

P(A^c) = 1 - P(A)

Addition rule

P(A \cup B) = P(A) + P(B) - P(A \cap B)

Mutually exclusive events

P(A \cup B) = P(A) + P(B)

Condition: when A \cap B = \varnothing

Monotonicity

A \subseteq B \implies P(A) \le P(B)

Applications

Modeling all possible outputs of a randomized algorithm as a sample space is the starting point for analyzing its expected running time or error probability.

Worked Examples

List all ordered outcomes for two coins.
$\Omega = \{HH, HT, TH, TT\}$
Identify which outcomes have at least one head.
$A = \{HH, HT, TH\}$

Answer: Ω = {HH, HT, TH, TT}; A = {HH, HT, TH}, so P(A) = 3/4.

Practice Problems

Difficulty 2/10

A single card is drawn from a standard 52-card deck. Describe the sample space and find P(the card is a heart).

Difficulty 3/10

If A and B are mutually exclusive events with P(A) = 0.3 and P(B) = 0.4, what is P(A ∪ B)?

0.120.70.11.0

Difficulty 4/10

A batch of 50 steel bolts contains 3 defective ones. If one bolt is drawn at random, what is the sample space and the probability of drawing a defective bolt?

Common Mistakes

Common Mistake

Treating an event and an outcome as the same thing.

An outcome is a single element of Ω; an event is a subset (possibly containing many outcomes, one outcome, or even zero outcomes).

Common Mistake

Applying P(A ∪ B) = P(A) + P(B) even when A and B overlap.

This double-counts the overlap. The general addition rule subtracts the intersection: P(A ∪ B) = P(A) + P(B) − P(A ∩ B).

Quiz

For two mutually exclusive events, P(A ∪ B) equals:

P(A)·P(B)P(A) + P(B)P(A) + P(B) − P(A∩B)1

The sample space of a random experiment is:

the most likely outcomethe set of ALL possible outcomesthe probability of successa single event

Summary

The sample space Ω is the set of all possible outcomes of a random experiment.
An event is any subset of the sample space.
P(Ω) = 1 and P(∅) = 0; probabilities are monotone with respect to subset inclusion.
The general addition rule is P(A ∪ B) = P(A) + P(B) − P(A ∩ B), which simplifies when A and B are mutually exclusive.
Sample spaces can be finite, countably infinite, or uncountable, depending on the experiment.

References

WebsiteWikipedia — Sample space